Abstract
In this paper we show that any smoothable complex projective variety, smooth in codimension two, with klt singularities and numerically trivial canonical class admits a finite cover, etale in codimension one, that decomposes as a product of an abelian variety, and singular analogues of irreducible Calabi-Yau and irreducible symplectic varieties.
Highlights
The Beauville-Bogomolov decomposition theorem asserts that any compact Kähler manifold with numerically trivial canonical bundle admits an étale cover that decomposes into a product of a torus, and irreducible, -connected Calabi-Yau, and symplectic manifolds
With the development of the minimal model program, it became clear that singularities arise as an inevitable part of higher dimensional life
If X is any complex projective manifold with Kodaira dimension κ(X) = 0, standard conjectures of the minimal model program predict the existence of a birational contraction X Xmin, where Xmin has terminal singularities and KXmin ≡ 0
Summary
The Beauville-Bogomolov decomposition theorem asserts that any compact Kähler manifold with numerically trivial canonical bundle admits an étale cover that decomposes into a product of a torus, and irreducible, -connected Calabi-Yau, and symplectic manifolds (see [Bea83]). If X admits a projective Q-Gorenstein smoothing, there exist a normal projective variety Y , a quasi-étale cover Y → X, and a projective smoothing Y → C of Y over an algebraic curve such that KY /C ∼Z 0. By [KM98, Def. 2.52 and Lemma 2.53], there exists a normal analytic variety Y and a finite cover γ : Y → X , étale over Xreg, such that KY /∆ ∼Z 0. We see that there exist a normal projective variety Y1 as well as a quasi-étale cover γ1 : Y1 → Y , and a projective smoothing Y1 → C1 of Y1 over an algebraic curve such that KY1/C1 ∼Z 0, completing the proof of the proposition
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